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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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Spectral decomposition of $A_{ij}=x_i+x_j$

Suppose we have a vector of real numbers $x_1,\ldots,x_n$, does anyone see an easy way to obtain the spectral decomposition of $n$-by-$n$ matrix $A$ defined below? $$A_{ij}=x_i+x_j$$ It seems to be rank-2 with eigenvector matrix being mostly zeros.
Yaroslav Bulatov
  • 5,327
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Does this matrix have signature 0?

Let $A$ be a skew-symmetric $n\times n$ matrix with coefficients in $\mathbb{R}$ or $\mathbb{C}$. Then consider the following $2n\times 2n$ matrix: $$\begin{pmatrix}0&A\\-A&0 \end{pmatrix}.$$ This matrix is symmetric because $A^t=-A$. Then is the…
SoYu
  • 307
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Find scalars of vectors that equals to zero

$x\cdot\begin{bmatrix}0\\-3\\1\end{bmatrix}$ + $y\cdot\begin{bmatrix}-5\\2\\-4\end{bmatrix}$ + $z\cdot\begin{bmatrix}-20\\-1\\-13\end{bmatrix}$ =$\begin{bmatrix}0\\0\\0\end{bmatrix}$ x = ？ y = ? z = ? RREF =…
YZY
  • 129
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3 answers

When does the kernel of a function equal the image?

When does the kernel of a function equal the image? Thanks in advance
proofy
  • 185
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4 answers

Do we need to check all 10 axioms to verify that some set, call it $V$, is a vector space?

The definition of the vector space from the book: Let $\bf u, v$ and $\bf w$ be vectors in the set $V$. The set $V$ is called a vector space if it satisfies the following 10 axioms. The vector addition $\bf u + v$ is also in the vector space…
Stokolos Ilya
  • 3,371
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When will $(\boldsymbol{A} \boldsymbol{B})^k = \boldsymbol{A}^k \boldsymbol{B} ^k = \boldsymbol{B} ^k \boldsymbol{A}^k$, $\forall k \geq 2$

I have known that when $\boldsymbol{A}\boldsymbol{B}= \boldsymbol{B}\boldsymbol{A}$, the equality $(\boldsymbol{A} \boldsymbol{B})^k = \boldsymbol{A}^k \boldsymbol{B} ^k = \boldsymbol{B} ^k \boldsymbol{A}^k$, $\forall k \geq 2$ holds. But the…
Kris Prokins
  • 321
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If $P$ is real orthogonal matrix with $\det P = -1$, prove that $-1$ is an eigenvalue of $P$.

If $P$ is real orthogonal matrix with $\det P = -1$, prove that $-1$ is an eigenvalue of $P$. can anyone help me please how can I solve this problem?
user67634
  • 119
4
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to show $(I_n+S)$ and $(I_n-S)$ are non-singular if $S$ is skew symmetric

If $S$ be a real skew-symmetric matrix of order $n$ , prove that the matrices $(I_n+S)$ and $(I_n-S)$ are both non-singular. can anyone help me please to solve this problem.thanks for your help.
damini
  • 219
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linear independence of unknown value

Guys I am given a set of vectors $(v_1,\dots,v_{10}), \mathbb{R}^{10}, n=10$, which are linearly independent. It doesn't give me the values of the vectors but tells me to find if these are independent or not. a. $v_1+v_2, v_2+v_3, v_3+v_4,…
D-Man
  • 593
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How to show certain things related to scalar products

Let $(V,\langle,\rangle)$ be a euclidean vector space and $u, v \in V$ two vectors. With $||x||=\sqrt{\langle x,x\rangle}$ the parallelogram law $||u+v||^{2} + ||u-v||^{2} = 2||u||^{2} + 2||w||^{2}$ is valid. (a)Prove the parallelogram law. (b)Let…
ghshtalt
  • 2,753
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2 answers

Proof that all $n\times n$ matrices that are nilpotent of order $n$ are similar.

Can someone give a proof that all $n\times n$ matrices that are nilpotent of order $n$ are similar? A matrix $A$ is called nilpotent if there exists some positive integer $k$ such that $A^k$ is the $0$-matrix. The order of nilpotency of a matrix…
Kishalay Sarkar
  • 3,467
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2 answers

Prove operator has eigenvalue with fundamental theorem of algebra

Every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue. Axler proves this as follows. Suppose $T: V\rightarrow V$ with $\text{dim}(V) = n > 0$. Choose $v \in V$ with $v \neq 0$. Then $v, Tv, T^2v,\dots,T^nv$ is not…
rorty
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Show that If a sequence is exact the dual sequence is exact

The book is Bosch Linear Algebra $p.79$ We look at at this sequence: $U\overset{f}\rightarrow V\overset{g}\rightarrow W$ where $f$ and $g$ is linear. The respective dual sequence is: $W^*\overset{g^*}{\rightarrow}V^*\overset{f^*}\rightarrow…
New2Math
  • 1,265
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Finding a linear equation

I cannot understand this question: "Find a linear equation (and parametrics) to $v$ where $v$ is perpendicular to the line segment of the extremes $(1,2,1)$ and $B$ $(1,8, -5)$, dividing it in half."
Jonathan Prates
  • 143
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Let $f$ be a polynomial with real coefficients and $A$ a symmetric matrix of $n\times n$ with elements in $\mathbb{R}$. Prove that $f(A)$ is symmetric

Let $f$ be a polynomial with real coefficients and $A$ a symmetric matrix of $n\times n$ with elements in $\mathbb{R}$. Prove that $f(A)$ is symmetric. Suppose $A$ is hermitian and that $f$ has complex coefficients. Is hermitian the matrix of…
user63192
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